Combinatorial Consequences Research Articles

The equivalence of a generalized Martin's axiom to a combinatorial principle

AbstractA generalized version of Martin's axiom, called BACH, is shown to be equivalent to one of its combinatorial consequences, a generalization of P(c).

A normal form for relational databases that is based on domains and keys

A new normal form for relational databases, called domain-key normal form (DK/NF), is defined. Also, formal definitions of insertion anomaly and deletion anomaly are presented. It is shown that a schema is in DK/NF if and only if it has no insertion or deletion anomalies. Unlike previously defined normal forms, DK/NF is not defined in terms of traditional dependencies (functional, multivalued, or join). Instead, it is defined in terms of the more primitive concepts of domain and key, along with the general concept of a “constraint.” We also consider how the definitions of traditional normal forms might be modified by taking into consideration, for the first time, the combinatorial consequences of bounded domain sizes. It is shown that after this modification, these traditional normal forms are all implied by DK/NF. In particular, if all domains are infinite, then these traditional normal forms are all implied by DK/NF.

Some spaces related to topological inequalities proven by the Erdős-Rado theorem

The Erdös-Rado theorem is very useful in proving cardinal inequalities in topology. It has been suggested that certain of these inequalities might be strengthened. We note that trees constructed by Jensen and Gregory using various extra axioms of set theory yield several counterexamples to these suggestions; for example, a space X , | X | = ω 2 , c ( X ) = ω 1 , χ ( X ) = ω X,|X| = {\omega _2},c(X) = {\omega _1},\chi (X) = \omega , answering a question of Hajnal and Juhász. We consider the apparently similar relation between | X | , e ( X ) |X|,e(X) , and d Ψ ( X ) d\Psi (X) of Ginsburg and Woods. Using combinatorial consequences of V = L V = L , we construct G δ {G_\delta } tree families, and establish that, assuming V = L V = L , an infinite cardinal κ \kappa is weakly compact iff d Ψ ( X ) > κ , e a ( X ) ⊂ κ imply | X | > κ {\text {iff}}\,d\Psi (X) > \kappa ,{e_a}(X) \subset \kappa {\text {imply}}|X| > \kappa . We consider products of countable chain condition spaces, and show that, using Cohen forcing that ( 2 ω {2^\omega } can be anything allowed by König’s theorem and there are spaces X , Y , c ( X ) = c ( Y ) = ω , c ( X × Y ) = 2 ω X,Y,c(X) = c(Y) = \omega ,c(X \times Y) = {2^\omega } ). A variation is a space W with the property c ( W n ) = ω n − 1 c({W^n}) = {\omega _{n - 1}} .

On partitions into stationary sets

We shall apply some of the results of Jensen [4] to deduce new combinatorial consequences of the axiom of constructibility, V = L. We shall show, among other things, that if V = L then for each cardinal λ there is a set A ⊆ λ such that neither A nor λ – A contain a closed set of type ω1. This is an extension of a result of Silver who proved it for λ = ω2, providing a partial answer to Problem 68 of Friedman [2].The main results of this paper were obtained independently by both authors.If λ is an ordinal, E is said to be Mahlo (or stationary) in λ, if λ – E does not contain a closed cofinal subset of λ.Consider the statements:(J1) There is a class E of limit ordinals and a sequence Cλ defined on singular limit ordinals λ such that(i) E ⋂ μ is Mahlo in μ for all regular > ω;(ii) Cλ is closed and unbounded in λ;(iii) if γ < is a limit point of Cλ, then γ is singular, γ ∉ E and Cγ = γ ⋂ Cλ.For each infinite cardinal κ:(J2,κ) There is a set E ⊂ κ+ and a sequence Cλ(Lim(λ), λ < κ+) such that(i) E is Mahlo in κ+;(ii) Cλ is closed and unbounded in λ;(iii) if cf(λ) < κ, then card Cλ < κ;(iv) if γ < λ is a limit point of Cλ then γ ∉ E and Cγ = ϣ ⋂ Cλ.